Optimal Extrapolation Bounds for Sparse Fourier Sums
arXiv:2607.10501
2026
Architecture
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper provides a sharp, separation-free stability law for extrapolating a signal represented by k arbitrary-frequency Fourier atoms from an observed interval to nearby points outside it. Its transferable asset is the endpoint growth rate exp(O(k sqrt(delta))), which is substantially milder than prior bounds and remains valid even when frequencies are arbitrarily close. This can be used to control Fourier-feature and implicit-neural-representation outputs outside the training domain, either by a bounded Fourier layer or by a theorem-shaped extrapolation regularizer. Since the theorem hides constants, implementations should calibrate conservative constants empirically and test both in-domain accuracy and out-of-domain overshoot.
Ideas from this paper
Unverified
2026
Add a differentiable regularizer to neural networks that learn sparse Fourier coefficients or trainable Fourier-feature frequencies. It penalizes predicted energy just outside the training interval when that energy exceeds the theorem-shaped envelope relative to observed in-domain L2 energy, discouraging cancellation patterns that fit the observed interval but explode nearby.
Useful6/10
Difficulty3/10
Novelty8/10
Unverified
2026
Replace an unconstrained Fourier-feature block in an implicit neural representation or coordinate MLP with a sparsity-aware layer whose output gain is normalized according to the distance outside the training interval. The normalization uses the paper's endpoint law, preventing a small in-domain Fourier representation from producing arbitrarily large outputs just beyond the observed coordinate range.
Useful6/10
Difficulty4/10
Novelty7/10